2.3A PRODUCT AND QUOTIENT RULE Goals: To differentiate a product To differentiate a quotient To find derivative of tangent, cotangent, secant and cosecant To find higher-order derivatives
- first•derivative of last + last •derivative of first
This rule can be extended to three products or more, see page 119 at the bottom of the page!
- First de-last + last de-first
EXAMPLE 1, P. 120 USING THE PRODUCT RULE
- Derivative of a product CANNOT, in general, be written as product of two derivatives.
- Find the derivative of h(x) = (2x – 3x 2 )(4x – 5)
- h’(x) = (2x – 3x 2 )(4) + (4x – 5)(2 – 6x)
- = 8x – 12x 2 + 8x – 24x 2 – 10 + 30x
- Not using rule but foiling first:
- h(x) = -12x 3 + 23x 2 – 10x which derives to same thing.
- In many cases you MUST use product rule!
Putting in factored form as much as possible allows us to find the places where a horizontal tangent occurs more easily (where y’ = 0)
- Find the derivative of y = 4x 3 sin x
- Find the derivative of y = 3xsinx – 4cosx
- (identify the products within the function!)
Use the product rule if __________________________. Use Constant rule if ____________________________. both factors are variable one of the factors is constant.
- Notice the underlined part is applying the product rule
Break out that cowboy accent! Low dehigh – high delow all over low squared! http://www.youtube.com/watch?v=DdV2UZV7AoA
In general, see if numerator factors, and keep denominator in a factored form Note liberal use of parentheses!
- Ex 4, p. 121 Using the quotient rule
- Warning – pay special attention to the subtraction required in the numerator – more points lost there than anywhere!
- Sometimes a rewrite is needed here too:
- Ex. 5 p. 122 Rewriting before differentiating
- Find an equation of the tangent line to f(x) at (-1, 1)
- and tangent line is y = 1
CAUTION: NOT EVERY QUOTIENT DESERVES THE QUOTIENT RULE!!!!
- Original Function Rewrite Derive Simplify
- p. 126 #2-12 ev, 13-41 eoo, 53,65,69